Let $(Q,g)$ be a Riemannian manifold and let $q_0,q_1\in Q$ be two points that are joined by a unique geodesic $\gamma$ (this holds in particular if $q_1$ belongs to a normal neighborhood of $q_0$). Suppose that $q_0=\gamma(t_0)$ and $q_1=\gamma(t_1)$, then we define Synge's World function as
\begin{equation}
\label{Synge world function formula}
\sigma(q_0,q_1):=\dfrac{1}{2}(t_1-t_0)\int_{t_0}^{t_1}g_{ij}\dot{\gamma}^i\dot{\gamma}^jdt,\tag{1}
\end{equation}
where $\gamma^i$ are the components of $\gamma$ in a chart that contains $q_0$ and $q_T$.
Consider the Hamiltonian
\begin{equation}
\label{geodesic Hamiltonian formula}
\begin{array}{rcl}
H:T^*Q & \rightarrow&\mathbb{R} \\
(q,p) & \mapsto&H(q,p):=\frac{1}{2}g^{ij}(q)p_ip_j.
\end{array}\tag{2}
\end{equation}
It can be shown that its flow $\phi^t_H$ is the geodesic flow, which means that the integral curves of the Hamiltonian vector field $X_H$ are the geodesic.
I've read that Synge's World function can be thought as a generating function for the geodesic flow, which means that
\begin{equation}
\label{eq 1}
p_0=-\dfrac{\partial\sigma}{\partial q_0}\qquad p_1=\dfrac{\partial\sigma}{\partial q_1},\tag{3}
\end{equation}
where
\begin{equation}
(q_1,p_1)=\phi^{t_1-t_0}_{H}(q_0,p_0).\tag{4}
\end{equation}
How to prove the relations (3)?
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Qmechanic
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Giulio Binosi
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I know that kind of proofs. But I don't know how to apply it to Synge's World function. Where is the link between the general case of Hamilton's principal function and the particular case of Synge's World function? – Giulio Binosi Sep 22 '20 at 09:09
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The trick is to keep everything in the Lagrangian formulation. – Qmechanic Sep 22 '20 at 09:38
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The Lagrangian associated to the geodesic Hamiltonian is $L(q,\dot{q})=\frac{1}{2}g_{ij}\dot{q}^i\dot{q}^j$? – Giulio Binosi Sep 22 '20 at 13:32
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$\uparrow$ Yes. – Qmechanic Sep 22 '20 at 14:10
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So that's where Synge's World function comes from, right? – Giulio Binosi Sep 22 '20 at 15:09
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$\uparrow$ Yes. – Qmechanic Sep 22 '20 at 15:11
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Thank you very much – Giulio Binosi Sep 22 '20 at 15:12